Several conditions are given for a stochastic process
X
(
t
)
X(t)
on
[
0
,
1
]
[0,1]
to have a local time which is continuous in its time parameter (for example, in the Gaussian case, the integrability of
[
E
(
X
(
t
)
−
X
(
s
)
)
2
]
−
1
/
2
{[E{(X(t) - X(s))^2}]^{ - 1/2}}
over the unit square). Furthermore, for any Borel function
F
F
on
[
0
,
1
]
[0,1]
with a continuous local time, the approximate limit of
|
F
(
s
)
−
F
(
t
)
|
/
|
s
−
t
|
|F(s) - F(t)|/|s - t|
as
s
→
t
s \to t
is infinite for a.e.
t
∈
[
0
,
1
]
t \in [0,1]
and
s
|
F
(
s
)
=
F
(
t
)
s|F(s) = F(t)
is uncountable for a.e.
t
∈
[
0
,
1
]
t \in [0,1]
.